Robert's Pi Math Puzzle Geocoin

To hopefully be logged by 314 cachers

Did you know . . .?

has a long and interesting history!

(Ancient history--More pi history--A novel way to compute pi--The symbol for pi--Having fun with pi)

That the ratio of the circumference to the diameter of a circle is constant (namely, pi) has been recognized for as long as we have written records.
A ratio of 3:1 appears in the following biblical verse:
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23; II Chronicles 4, 2.)
The ancient Babylonians generally calculated the area of a circle by taking 3 times the square of its radius ( =3), but one Old Babylonian tablet (from ca. 1900-1680 BCE) indicates a value of 3.125 for pi.
Ancient Egyptians calculated the area of a circle by the following formula (where d is the diameter of the circle):

This yields an approximate value of 3.1605 for pi.

The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 < < 22/7. Archimedes's results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed.

Beginning with a hexagon, he worked all the way up to a ploygon with 96 sides!

Archimedes's method for approximating the value of pi.
(Source: http://www.math.psu.edu/dna/graphics.html#archimedes)
The approximate area of the circle lies between the areas of the circumscribed and the inscribed hexagons.

More pi history:

European mathematicians in the early modern period developed new arithmetical formulae to approximate the value of pi, such as that of James Gregory (1638-1675), which was taken up by Leibniz:

One problem with using this formula to calculate the value of pi is that you would have to add 5 million terms to work out a value of /4 that extends to 6 or 7 decimal places!
In 1706, another mathematician named John Machin developed a refinement on Gregory's formula, yielding the formula still used today by computer programmers to compute pi:
Using this formula, an Englishman named William Shanks calculated pi to 707 places, a labor of many years, which he published in 1873. (Only 527 places were correct, however!)

A novel way to compute pi:

An eighteenth-century French mathematician named Georges Buffon devised a way to calculate pi based on probability. Buffon's method begins with a uniform grid of parallel lines, a unit distance apart. If you drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/ . Various people have tried to calculate pi by throwing needles. Depending on when you stop the experiment, you can obtain a reasonably accurate estimate of pi. You can try Buffon's needle experiment for yourself (virtually) at http://www.angelfire.com/wa/hurben/buff.html Some ants apparently actually use this algorithm to measure the size of potential nest sites!

The symbol for pi:

was introduced by the English mathematician William Jones in 1706, who wrote: 3.14159 =
This symbol was adopted by Euler in 1737 and became the standard symbol for pi.

Having fun with pi:

Some people are just crazy about pi!
There are pi poems . . .
There are pieces of music based on the digits of pi . . .
There is a web site where you can find your birthday in pi . . .
There are people who have memorized 1000+ digits of pi . . .
In 1897 the Indiana Legislature tried to legally establish the value of pi . . .
Pi has earned a spot in "The Useless Pages" .